Dice Similarity Measure between Single Valued Neutrosophic Multisets and Its Application in Medical. Diagnosis

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1 Neutrosophc Sets ad Systems, Vol. 6, Dce Smlarty Measure betwee Sgle Valued Neutrosophc Multsets ad ts pplcato Medcal Dagoss Sha Ye ad Ju Ye Tasha Commuty Health Servce Ceter. 9 Hur rdge, Yuecheg Dstrct, Shaoxg, Zheag 3000, P.R. Cha. E-mal: shayeh@sa.com Departmet of Electrcal ad formato Egeerg, Shaoxg Uversty, 508 Huacheg West Road, Shaoxg, Zheag 3000, P.R. Cha. E-mal: yehu@alyu.com bstract. Ths paper troduces the cocept of a sgle valued eutrosophc multset (SVNM as a geeralzato of a tutostc fuzzy multset (FM ad some basc operatoal relatos of SVNMs, ad the proposes the Dce smlarty measure ad the weghted Dce smlarty measure for SVNMs ad vestgates ther propertes. Fally, the Dce smlarty measure s appled to a medcal dagoss problem wth SVNM formato. Ths dagoss method ca deal wth the medcal dagoss problem wth determate ad cosstet formato whch caot be hadled by the dagoss method based o FMs. Keywords: Sgle valued eutrosophc set, multset, sgle valued eutrosophc multset, Dce smlarty measure, medcal dagoss. troducto medcal dagoss problems, physcas ca obta a lot of formato from moder medcal techologes, whch s ofte complete ad determate formato due to the complexty of varous dseases. Therefore, real medcal dagoss cotas lots of complete ad ucertaty formato, whch s a usual pheomeo of medcal dagoss problems. To represet complete ad ucertaty formato, taassov [] troduced tutostc fuzzy sets (FSs as a geeralzato of fuzzy sets []. The promet characterstc of FS s that a membershp degree ad a o-membershp degree are assged to each elemet the set. The, varous medcal dagoss methods have bee preseted uder tutostc fuzzy evromets [3, 4]. Recetly, Ye [5] proposed a cose smlarty measure betwee FSs ad appled t to patter recogto ad medcal dagoss. Hug [6] troduced a tutostc fuzzy lkelhood-based measuremet ad appled t to the medcal dagoss ad bactera classfcato problems. Further, Ta [7] developed the cotaget smlarty measure of FSs ad appled t to medcal dagoss. s a geeralzato of fuzzy sets ad FSs, Wag et al. [8] troduced a sgle valued eutrosophc set (SVNS as a subclass of the eutrosophc set proposed by Smaradache [9]. SVNS cossts of the three terms lke the truth-membershp, determacy-membershp ad falstymembershp fuctos ad ca be better to express determate ad cosstet formato, but fuzzy sets ad FSs caot hadle determate ad cosstet formato. However, smlarty measures play a mportat role the aalyss ad research of medcal dagoss, patter recogto, mache learg, decso makg, ad clusterg aalyss ucertaty evromet. Therefore, varous smlarty measures of SVNSs have bee proposed ad maly appled them to decso makg ad clusterg aalyss. For stace, Maumdar ad Samata [0] troduced several smlarty measures of SVNSs based o dstaces, a matchg fucto, membershp grades, ad the proposed a etropy measure for a SVNS. Ye [] proposed three vector smlarty measures for smplfed eutrosophc sets (SNSs, cludg the Jaccard, Dce, ad cose smlarty measures for SVNSs ad terval eutrosophc sets (NSs, ad appled them to multcrtera decso-makg problems wth smplfed eutrosophc formato. Ye [] ad Ye ad Zhag [3] further proposed the smlarty measures of SVNSs for decso makg problems. Furthermore, Ye [4] put forward dstacebased smlarty measures of SVNSs ad appled them to clusterg aalyss. real medcal dagoss problems, however, by oly takg oe tme specto, we woder whether oe ca obta a cocluso from a partcular perso wth a partcular decease or ot. Sometmes he/she may also show the symptoms of dfferet dseases. The, how ca we gve a proper cocluso? Oe soluto s to exame the patet at dfferet tme tervals (e.g. two or three tmes a day. ths case, a fuzzy multset cocept troduced by Yager [5] s very sutable for expressg ths formato at dfferet tme tervals, whch allows the repeated occurreces of ay elemet. Thus, the fuzzy multset ca occur more tha oce wth the possblty of the same or dfferet membershp values. The, Sho ad Sul [6] exteded the fuzzy multset to the tutostc fuzzy multset (FM ad preseted some basc operatos ad a dstace measure for FMs, ad the appled the dstace measure to Sha Ye, Ju Ye, Dce Smlarty Measure betwee Sgle Valued Neutrosophc Multsets ad ts pplcato Medcal Dagoss

2 49 Neutrosophc Sets ad Systems, Vol. 6, 04 medcal dagoss problem. Raaraeswar ad Uma [7] preseted the Hammg dstace-based smlarty measure for FMs ad ts applcato medcal dagoss. However, exstg FMs caot represet ad deal wth the determacy ad cosstet formato whch exsts real stuatos (e.g. medce dagoss problems. To hadle the medcal dagoss problems wth determacy ad cosstet formato, the ams of ths paper are: ( to troduce a sgle valued eutrosophc multset (SVNM as a geeralzato of FMs ad some operatoal relatos for SVNMs, ( to propose the Dce smlarty measure of SVNMs, (3 to apply the Dce smlarty measure to medcal dagoss. The rest of the artcle s orgazed as follows. Secto troduces some basc cocepts of FSs, FMs, ad SVNSs. Sectos 3 troduces a cocept of SVNM ad some operatoal relatos of SVNMs. Secto 4, we preset the Dce smlarty measure ad the weghted Dce smlarty measure for SVNMs ad vestgate ther propertes. Secto 5, we apply the proposed smlarty measure to a medcal dagoss problem. Coclusos ad further research are cotaed Secto 6. Prelmares. Some basc cocepts of FSs ad FMs taassov [] troduced FSs as a exteso of fuzzy sets [] ad gave the followg defto. Defto []. FS the uverse of dscourse X s defed as { (, ( x X}, where (: X [0, ] ad (: X [0, ] are the membershp degree ad o-membershp degree of the elemet x to the set wth the codto 0 ( + ( for x X. The, ( = ( ( s called taassov's tutostc dex or a hestacy degree of the elemet x the set. obvously there s 0 ( for x X. Further, Sho ad Sul [6] troduced a FM cocept by combg the two cocepts for FSs ad fuzzy multsets together ad gave the followg defto. Defto [6]. Let X be a oempty set. The, a FM draw from X s characterzed by the two fuctos: cout membershp of CM ad cout o-membershp of CN such that CM (: X R ad CN (: X R for x X, where R s the set of all real umber multsets draw from the ut terval [0, ]. Thus, a FM s deoted by ( (, (,..., (,( (, (,..., (, x X where the membershp seuece ( (, (,..., ( s a decreasgly ordered seuece ( (,..., (, the correspodg o- membershp seuece ( (, (,..., ( may ot be decreasg or creasg order, ad the sum of ( ad ( satsfes the codto 0 ( + ( for x X ad =,,,. For coveece, a FM ca be deoted by the followg smplfed form: (, ( x X,,,...,. Let (, ( x X,,,..., (, ( x X,,,..., ad be two FMs. The there are the followg relatos [6]: c ( Complemet: (, ( x X,,,..., ( cluso: f ad oly f ( (, ( ( for =,,, ad x X (3 Eualty: = f ad oly f ad (4 Uo: ( (, ( ( x X,,,..., (5 tersecto: ( (, ( ( x X,,,..., (6 ddto: ( ( ( (, ( ( x X,,,..., (7 Multplcato: ( (, ( ( ( (,. x X,,,...,. Some cocepts of SVNSs Smaradache [9] orgally preseted the cocept of a eutrosophc set from phlosophcal pot of vew. eutrosophc set a uversal set X s characterzed by a truth-membershp fucto T (, a determacymembershp fucto (, ad a falsty-membershp fucto F (. The fuctos T (, (, F ( X are real stadard or ostadard subsets of ] 0, + [, such that T (: X ] 0, + [, (: X ] 0, + [, ad F (: X ] 0, + [. The, the sum of T (, ( ad F ( satsfes 0 sup T ( + sup ( + sup F ( 3 +. However, the eutrosophc set troduced from phlosophcal pot of vew s dffcult to apply t to practcal applcatos. Thus, Wag et al. [8] troduced a SVNS as a subclass of the eutrosophc set ad the followg defto of SVNS. Sha Ye, Ju Ye, Dce Smlarty Measure betwee Sgle Valued Neutrosophc Multsets ad ts pplcato Medcal Dagoss

3 Neutrosophc Sets ad Systems, Vol. 6, Defto 3 [8]. Let X be a uversal set. SVNS X s characterzed by a truth-membershp fucto T (, a determacy-membershp fucto (, ad a falstymembershp fucto F (. The, a SVNS ca be deoted as T (, (, F ( x X, where the sum of T (, (, F ( [0, ] satsfes 0 T ( + ( + F ( 3 for each x X. T (, (, F ( x X ad For two SVNSs x T (, (, F ( x X,, there are the followg relatos [8]: c ( Complemet: x F (, (, T ( x X, ( cluso: f ad oly f T ( T (, ( (, F ( F ( for ay x X (3 Eualty: = f ad oly f ad (4 Uo: T ( T (, ( (, F ( F ( x X (5 tersecto:. T ( T (, ( (, F ( F ( x X 3 Sgle valued eutrosophc multsets Ths secto troduces SVNMs as a geeralzato of SVNSs ad FMs ad some operatoal relatos for SVNMs. Defto 4. Let X be a oempty set wth geerc elemets X deoted by x. SVNM draw from X s characterzed by the three fuctos: cout truthmembershp of CT, cout determacy-membershp of C, ad cout falsty-membershp of CF such that CT (: X R, C (: X R, CF (: X R for x X, where R s the set of all real umber multsets the real ut terval [0, ]. The, a SVNM s deoted by ( T(, T (,..., T (, ( (, (, ( F(, F (, F ( (,..., x X, where the truth-membershp seuece ( T (, T (,..., T (, the determacy-membershp seuece ( (, (,..., (, ad the falstymembershp seuece ( F (, F (,..., F ( may be decreasg or creasg order, ad the sum of T (, (, F ( [0, ] satsfes the codto 0 sup ( + sup ( + sup F ( 3 for x X ad =,,,. T For coveece, a SVNM ca be deoted by the smplfed form: T (, (, F ( x X,,,...,. Defto 5. The legth of a elemet x a SVNM s defed as the cardalty of CT ( or C (, or CF ( ad s deoted by L(x:. The L(x: = CT ( = C ( = CF (. Defto 6. Let ad be two SVNMs X, the the legth of a elemet x ad s deoted by l x = L(x:, = max{l(x:, L(x: }. For example, we cosder SVNMs the set X = {x, x, x 3 } as = {<x, (0., 0., (0., 0.3, (0.6, 0.8>, < x, (0.3, 0.4, 0.5, (0., 0.3, 0.4, (0.5, 0.6, 0.7>}, = {<x, (0., (0., (0.4 >, < x 3, (0.3, 0.4, 0.5, 0.6, (0., 0., 0.3, 0.4, (0., 0., 0.3, 0.5>}. Thus, there are L(x : =, L(x : = 3, L(x 3 : = 0 L(x : =, L(x : = 0, L(x 3 : = 4, l x = L(x :, =, l x = L(x :, = 3, ad l x3 = L(x 3 :, = 4. For coveet operato betwee SVNMs ad X, oe ca make L(x: = L(x: by appedg suffcet mmal umbers for the truth-membershp degree ad suffcet maxmum umbers for the determacymembershp ad falsty-membershp degrees as pessmsts or suffcet maxmum umbers for the truth-membershp value ad suffcet mmal umbers for the determacy-membershp ad falsty-membershp values as optmsts. Defto 7. Let = { T (, (, F( x X, =,,, } ad = { T (, (, F ( x X, =,,, } be two SVNMs X. The, there are the followg relatos: ( cluso: f ad oly f T ( T (, ( (, F ( F ( for =,,, ad x X ( Eualty: = f ad oly f ad (3 Complemet: c F (, (, T ( x X,,,..., (4 Uo: T ( T (, ( (, F ( F ( x X,,,..., (5 tersecto: T ( T (, ( (, F( F (. x X,,,..., Sha Ye, Ju Ye, Dce Smlarty Measure betwee Sgle Valued Neutrosophc Multsets ad ts pplcato Medcal Dagoss

4 5 Neutrosophc Sets ad Systems, Vol. 6, 04 4 Dce smlarty measure of SVNMs ths secto, we propose the Dce smlarty measure ad the weghted Dce smlarty measure for SVNMs ad vestgate ther propertes. Defto 8. Let = {x, T x,, F x X, = (,,, } ad = {x, T x,, F x X, = (,,, } be ay two SVNMs X = {x, x,, x }. The, we defe the followg Dce smlarty measure betwee ad : SD (, l l l l l l T F T T F( x F T F, ( where l = L(x :, = max{l(x :, L(x : } for =,,,. The, the Dce smlarty measure has the followg Proposto : Proposto. For two SVNMs ad X = {x, x,, x }, the Dce smlarty measure S D (, should satsfy the followg propertes (P-(P3: Proof: (P 0 S D (, (P S D (, = S D (, (P3 S D (, = f =,.e., T x = T x, ( ( x = x, F x = F x for ( ( ( ( every x X, =,,,, ad =,,...,. (P t s obvous that the property s true accordg to the eualty a b ab for E. (. (P t s straghtforward. (P3 f =, the there are T x = T x, x = ( ( ( x, F x = F x for every x X, =,,, ( ( ( ad =,,...,. Hece there s S D (, =. Takg the weght w of each elemet x ( =,,, to accout wth w [0, ] ad w, we troduce the followg weghted Dce smlarty measure betwee SVNMs ad : W (, T F Sha Ye, Ju Ye, Dce Smlarty Measure betwee Sgle Valued Neutrosophc Multsets ad ts pplcato Medcal Dagoss D l w l l l l l T T F( x F T F, ( where l = L(x :, = max{l(x :, L(x : } for =,,,. f W = (/, /,, / T, the E. ( reduces to E. (. The, the weghted Dce smlarty measure has the followg Proposto : Proposto. For two SVNMs ad X = {x, x,, x }, the weghted Dce smlarty measure W D (, should satsfy the followg propertes (P-(P3: (P 0 W D (, (P W D (, = W D (, (P3 W D (, = f =,.e., T x = T x, ( ( x = x, F x = F x for every x X, ( ( ( ( =,,, ad =,,...,. y a smlar proof method of Proposto, we ca prove that the propertes (P (P3. 5 Medcal dagoss usg the Dce smlarty measure ths secto, we apply the Dce smlarty measure to the medcal dagoss problem wth SVNM formato. The detals of a typcal example adapted from [6] are gve below. Let P = {P, P, P 3, P 4 } be a set of four patets, D = {D, D, D 3, D 4 } = {Vral fever, Tuberculoss, Typhod, Throat dsease} be a set of dseases, ad S = {S, S, S 3, S 4, S 5 } = {Temperature, Cough, Throat pa, Headache, ody pa} be a set of symptoms. the medcal dagoss problem, whe we have to take three dfferet samples three dfferet tmes a day (e.g. morg, oo ad ght, the characterstc values betwee patets ad the dcated symptoms are represeted by the followg SVNMs: P ={<S, (0.8, 0.6, 0.5, (0.3, 0., 0., (0.4, 0., 0.>, <S, (0.5, 0.4, 0.3, (0.4, 0.4, 0.3, (0.6, 0.3, 0.4>, <S 3, (0., 0., 0.0, (0.3, 0., 0., (0.8, 0.7, 0.7>, <S 4, (0.7, 0.6, 0.5, (0.3, 0., 0., (0.4, 0.3, 0.>, <S 5, (0.4, 0.3, 0., (0.6, 0.5, 0.5, (0.6, 0.4, 0.4>} P ={<S, (0.5, 0.4, 0.3, (0.3, 0.3, 0.,(0.5, 0.4, 0.4>, <S, (0.9, 0.8, 0.7, (0., 0., 0., (0., 0., 0.0>, <S 3, (0.6, 0.5, 0.4, (0.3, 0., 0., (0.4, 0.3, 0.3>, <S 4, (0.6, 0.4, 0.3, (0.3, 0., 0., (0.7, 0.7, 0.3>, <S 5, (0.8, 0.7, 0.5, (0.4, 0.3, 0., (0.3, 0., 0.>

5 Neutrosophc Sets ad Systems, Vol. 6, 04 5 P 3 ={<S, (0., 0., 0., (0.3, 0., 0., (0.8, 0.7, 0.6>, <S, (0.3, 0., 0., (0.4, 0., 0., (0.7, 0.6, 0.5>, <S 3, (0.8, 0.8, 0.7, (0., 0., 0., (0., 0., 0.0>, <S 4, (0.3, 0., 0., (0.3, 0.3, 0.3, (0.7, 0.6, 0.6>, <S 5, (0.4, 0.4, 0.3, (0.4, 0.3, 0., (0.7, 0.7, 0.5> P 4 ={<S, (0.5, 0.5, 0.4, (0.3, 0., 0., (0.4, 0.4, 0.3>, <S, (0.4, 0.3, 0., (0.4, 0.3, 0., (0.7, 0.5, 0.3>, <S 3, (0.7, 0., 0.0, (0.4, 0.3, 0.3, (0.7, 0.7, 0.6>, <S 4, (0.6, 0.5, 0.3, (0.6, 0., 0., (0.6, 0.4, 0.3>, <S 5, (0.5, 0., 0., (0.3, 0.3, 0., (0.6, 0.5, 0.4>. The, the characterstc values betwee symptoms ad the cosdered dseases are represeted by the form of SVNSs: D (Vral fever = {<S, 0.8, 0., 0.>, <S, 0., 0.7, 0.>, <S 3, 0.3, 0.5, 0.>, <S 4, 0.5, 0.3, 0.>, <S 5, 0.5, 0.4, 0.>} D (Tuberculoss = {<S, 0., 0.7, 0.>, <S, 0.9, 0.0, 0.>, <S 3, 0.7, 0., 0.>, <S 4, 0.6, 0.3, 0.>, <S 5, 0.7, 0., 0.>} D 3 (Typhod = {<S, 0.5, 0.3, 0.>, <S, 0.3, 0.5, 0.>, <S 3, 0., 0.7, 0.>, <S 4, 0., 0.6, 0.>, <S 5, 0.4, 0.4, 0.>} D 4 (Throat dsease = {<S, 0., 0.7, 0.>, <S, 0.3, 0.6, 0.>, <S 3, 0.8, 0., 0.>, <S 4, 0., 0.8, 0.>, <S 5, 0., 0.8, 0.>}. The, by usg E. (, we ca obta the Dce smlarty measure betwee each patet P ( =,, 3, 4 ad the cosdered dsease D ( =,, 3, 4, whch are show Table. D (Vral fever Table Measure values of S D (P, D D (Tuberculoss D 3 (Typhod D 4 (Throat dsease P P P P Tables, the largest smlarty measure dcates the proper dagoss. Hece, Patet P suffers from typhod, Patet P suffers from vral fever, Patet P 3 also suffers from typhod, ad Patet P 4 suffers from tuberculoss. 6 Cocluso Ths paper troduced a cocept of SVNM ad some basc operatoal relatos of SVNMs, ad the proposed the Dce smlarty measure ad the weghted Dce smlarty measure for SVNMs ad vestgated ther propertes. Fally, the Dce smlarty measure of SVNMs was appled to medce dagoss uder the SVNM evromet. The Dce smlarty measure of SVNMs s effectve hadlg the medcal dagoss problems wth determate ad cosstet formato whch the smlarty measures of FMSs caot hadle, because FMSs caot express ad deal wth determate ad cosstet formato. further work, t s ecessary ad meagful to exted SVNMs to terval eutrosophc multsets ad ther operatos ad measures ad to vestgate ther applcatos such as decso makg, patter recogto, ad medcal dagoss. Refereces [] K. taassov. tutostc fuzzy sets. Fuzzy Sets ad Systems, 0 (986, [] L.. Zadeh, Fuzzy Sets. formato ad Cotrol, 8 (965, [3] S. K De, R swas, ad. R. Roy. applcato of tutostc fuzzy sets medcal dagoss. Fuzzy Sets ad Systems, 7( (00, [4]. K. Vlachos ad G. D. Sergads. tutostc fuzzy formato pplcatos to patter recogto. Patter Recogto Letters, 8 (007, [5] J. Ye. Cose smlarty measures for tutostc fuzzy sets ad ther applcatos. Mathematcal ad Computer Modellg, 53(- (0, [6] K. C. Hug. pplcatos of medcal formato: Usg a ehaced lkelhood measured approach based o tutostc fuzzy sets. E Trasactos o Healthcare Systems Egeerg, (3 (0, 4-3. [7] M. Y. Ta. ew fuzzy smlarty based o cotaget fucto for medcal dagoss. dvaced Modelg ad Optmzato, 5( (03, [8] H. Wag, F. Smaradache, Y. Q. Zhag, ad R. Suderrama. Sgle valued eutrosophc sets. Multspace ad Multstructure, 4 (00, [9] F. Smaradache. ufyg feld logcs. eutrosophy: Neutrosophc probablty, set ad logc. Rehoboth: merca Research Press, 999. [0] P. Maumdar ad S. K. Samata. O smlarty ad etropy of eutrosophc sets. Joural of tellget ad Fuzzy Systems, 6(3 (04, [] J. Ye. Vector smlarty measures of smplfed eutrosophc sets ad ther applcato multcrtera decso makg. teratoal Joural of Fuzzy Systems, 6( (04, 04-. [] J. Ye. Multple attrbute group decso-makg method wth completely ukow weghts based o smlarty measures uder sgle valued eutrosophc evromet. Joural of tellget ad Fuzzy Systems, (04, do: 0.333/FS-45. [3] J. Ye ad Q. S. Zhag, Sgle valued eutrosophc smlarty measures for multple attrbute decso makg. Neutrosophc Sets ad Systems (04, [4] J. Ye. Clusterg methods usg dstace-based smlarty measures of sgle-valued eutrosophc sets. Joural of tellget Systems, (04, do: 0.55/sys [5] R. R. Yager. O the theory of bags, (Mult sets. teratoal Joural of Geeral System, 3 (986, [6] T. K. Sho ad J. J. Sul. tutostc fuzzy mult sets ad ts applcato medcal dagoss. World cademy of Scece, Egeerg ad Techology, 6( (0, 48- Sha Ye, Ju Ye, Dce Smlarty Measure betwee Sgle Valued Neutrosophc Multsets ad ts pplcato Medcal Dagoss

6 53 Neutrosophc Sets ad Systems, Vol. 6, [7] P. Raaraeswar ad N. Uma. Normalzed hammg smlarty measure for tutostc fuzzy mult sets ad ts applcato medcal dagoss. teratoal Joural of Mathematcs Treds ad Techology, 5(3 (04, 9-5. Receved: September, 04. ccepted: October 0, 04. Sha Ye, Ju Ye, Dce Smlarty Measure betwee Sgle Valued Neutrosophc Multsets ad ts pplcato Medcal Dagoss